Download Unit 4.7 Notes: Concurrent Lines, Medians, and Altitudes

Unit 4.7 Notes: Concurrent Lines, Medians, and Altitudes
Angle Bisectors: Construct the angle bisectors for each of the three angles in the following triangles:
Definition: Concurrent – When three or more lines intersect at one point.
Do the angle bisectors you constructed above have a point of concurrency in each of your triangles?
any such points as point I for each of your triangles above.
Label
Make a conjecture about the angle bisectors of a triangle:
Concurrency of Angle Bisectors Theorem
Perpendicular Bisectors: Now, construct perpendicular bisectors for each of the three sides of the triangles below:
Do the perpendicular bisectors you constructed above have a point of concurrency in each of your triangles?
Label any such points as point C for each of your triangles above.
Concurrency of Perpendicular Bisectors Theorem
Definition: Circumcenter
of a Triangle– The point of concurrency of the perpendicular bisectors of a
triangle.
A circle can be circumscribed bout the triangle with the circumcenter as the middle point. (The vertices of the
triangle will be on the edges of the circle)
Definition: Incenter
of a Triangle– The point of concurrency of the angle bisectors of a triangle.
A circle can be inscribed in the triangle with the incenter as the middle point.
Definition: Median
of a Triangle– The segment whose endpoints are a vertex and the midpoint of the
opposite side.
Construct the three medians for each of the following triangles. Label the point of concurrency as point C for each:
Medians of a Triangle Theorem
Example:
B
A
C
Definition: Centroid
of a Triangle– The point of concurrency of the medians of a triangle.
Definition: Altitude
of a Triangle– The perpendicular segment from a vertex to the line containing the
opposite side.
Construct the three altitudes for each of the following triangles. Label the point of concurrency as point O for each:
Concurrency of Altitudes of a Triangle Theorem
Definition: Orthocenter– The point of concurrency of the altitudes of a triangle.
4.7 Summary of New Words: